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  "body": "\n## Example\n\nLet $d \\ge 2$, let $\\lambda_1, \\dots, \\lambda_d \u003e 0$ be lengths with sum $\\abs{\\lambda} = \\lambda_1 + \\cdots + \\lambda_d$, and let $\\pi$ be a permutation of $\\set{1, \\dots, d}$.\nCut $I = [0, \\abs{\\lambda})$ into the intervals $I_j = [\\lambda_1 + \\cdots + \\lambda_{j-1}, \\lambda_1 + \\cdots + \\lambda_j)$, for $j = 1, \\dots, d$.\nThe *interval exchange transformation* $T \\colon I \\to I$ translates each $I_j$ so that the intervals appear in the new order $I_{\\pi^{-1}(1)}, I_{\\pi^{-1}(2)}, \\dots, I_{\\pi^{-1}(d)}$ from left to right.\nExplicitly, for $x \\in I_j$,\n$$T(x) = x - \\sum_{i \u003c j} \\lambda_i + \\sum_{\\pi(i) \u003c \\pi(j)} \\lambda_i.$$\n\nCheck that $T$ is a [[measure-preserving transformation|556WBDAHNP]] for [[Lebesgue measure|2Z2QVWZ262]].\nIndeed, $T$ is a bijection of $I$, and it is a translation on each of the finitely many intervals $I_j$.\nTranslations preserve length, so for every Borel set $E \\subseteq I$, $\\abs{T^{-1}E} = \\sum_j \\abs{T^{-1}E \\cap I_j} = \\sum_j \\abs{E \\cap T(I_j)} = \\abs{E}$.\n\nConcrete case: $d = 2$ and $\\pi$ the transposition, with $0 \u003c \\alpha \u003c 1$ and lengths $\\lambda_1 = 1 - \\alpha$ and $\\lambda_2 = \\alpha$.\nThen $T(x) = x + \\alpha$ on $[0, 1 - \\alpha)$ and $T(x) = x - (1 - \\alpha)$ on $[1 - \\alpha, 1)$, so $T(x) = x + \\alpha \\bmod 1$, the [[rotation of the circle|2Q1J7RZ4TK]].\nInterval exchanges are thus a generalization of circle rotations.\n\nFor an irreducible permutation $\\pi$ and Lebesgue almost every length vector, the interval exchange is [[uniquely ergodic|NZZE9KCAD8]].\nMasur and Veech proved this independently in 1982.\n\n## Explanations\n\n### L1 human\n\nCut a stick into several pieces, shuffle the pieces into a new order without flipping them, and glue them back.\nThat rearrangement is an interval exchange.\nRepeating it moves points around in a way that never changes the total length of any set.\nWith two pieces it is just a rotation of a circle; with more pieces it already leads to deep questions, linked to flat surfaces and billiards in polygons.\n",
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